Theorem wleoa 376

Description: Relation between two methods of expressing "less than or equal to". (Contributed by NM, 27-Sep-1997.)

Hypothesis

Ref	Expression
wleoa.1	((a ∪ c) ≡ b) = 1

Assertion

Ref	Expression
wleoa	((a ∩ b) ≡ a) = 1

Proof of Theorem wleoa

Step	Hyp	Ref	Expression
1		wleoa.1	. . . 4 ((a ∪ c) ≡ b) = 1
2	1	wr1 197	. . 3 (b ≡ (a ∪ c)) = 1
3	2	wlan 370	. 2 ((a ∩ b) ≡ (a ∩ (a ∪ c))) = 1
4		wa5c 201	. 2 ((a ∩ (a ∪ c)) ≡ a) = 1
5	3, 4	wr2 371	1 ((a ∩ b) ≡ a) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  wdf2le2  386  wdid0id5  1111  wdid0id1  1112  wdid0id2  1113  wdid0id3  1114  wdid0id4  1115